Question

A developer has land that has x feet of lake frontage. The land is to be subdivided into lots, each of which is to have either 80 feet or 100 feet of Lake Frontage. If 1/9 of the lots are to have 80 feet of frontage each and the remaining 40 lots are to have 100 feet of frontage each, what is the value of x?
$\left( a \right)400$
$\left( b \right)3200$
$\left( c \right)3700$
$\left( d \right)4400$
$\left( e \right)4760$

Answer 1

Hint: In this particular type of question use the concept that x feet of Lake Frontage is equal to the sum of the product of (80 and number of 80 feet long lots) and (100 and the number of 100 feet long lots), so if we find out the number of 80 and 100 feet long lots we will easily find the value of x, so use these concepts to reach the solution of the question.

Complete step-by-step solution:
A developer has land that has x feet of lake frontage.
Now according to the question, he divided his x feet of Lake Frontage into lots of 80 feet long and 100 feet long lots respectively.
So x feet of Lake Frontage is equal to the sum of the number of 80 feet long lots and the number of 100 feet long lots.
Let the number of 80 feet long lots = a,
And the number of 100 feet long lots = b.
So, x feet of lake frontage = 80a + 100b
$ \Rightarrow x = 80a + 100b$..................... (1)
Now it is given that 1/9 of the lots are to have 80 feet of frontage each and the remaining 40 lots are to have 100 feet of frontage each.
So the number of 100 feet of frontage = 40.
Therefore, b = 40
Let the total lots be y
Therefore, y = a + b............ (2)
So, $\dfrac{1}{9}$ of y is 80 feet of frontage and $\left( {y - \dfrac{y}{9}} \right) = \dfrac{{8y}}{9}$ is 100 feet of frontage
Now 100 feet of frontage is 40.
$ \Rightarrow \dfrac{{8y}}{9} = 40$
$ \Rightarrow y = 45$
Now from equation (2) we have,
$ \Rightarrow y = a + b$
$ \Rightarrow 45 = a + 40$
$ \Rightarrow a = 5$
So there are 5 lots which are 80 feet long.
Now substitute the values of a and b in equation (1) we have,
\[ \Rightarrow x = 80\left( 5 \right) + 100\left( {40} \right) = 400 + 4000 = 4400\]
So, the man has 4400 feet of Lake Frontage.
So this is the required answer.
Hence option (d) is the correct answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall that the number of 100 feet long lots are given i.e. 40 so if the total number of lots is y so the number of 80 feet long lots are, y – 40, so $\dfrac{1}{9}$ of y is 80 feet long lots and remaining lots i.e. $\left( {y - \dfrac{y}{9}} \right) = \dfrac{{8y}}{9}$ are the 100 feet long lots so we easily calculate the total number of lots and the 80 feet long lots.